According To The Fundamental Theorem Of Algebra, How Many Roots Exist For The Polynomial Function F ( X ) = ( X 3 − 3 X + 1 ) 2 F(x)=\left(x^3-3 X+1\right)^2 F ( X ) = ( X 3 − 3 X + 1 ) 2 ?A. 2 Roots B. 3 Roots C. 6 Roots D. 9 Roots

Mar 10, 2025 by ADMIN 236 views

**Understanding the Fundamental Theorem of Algebra and Polynomial Functions**

The Fundamental Theorem of Algebra is a fundamental concept in mathematics that states every non-constant polynomial equation of degree n has exactly n complex roots. This theorem is crucial in understanding the behavior of polynomial functions and their roots. In this article, we will explore the concept of the Fundamental Theorem of Algebra and apply it to a given polynomial function to determine the number of roots it has.

What is the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra is a theorem in algebra that states every non-constant polynomial equation of degree n has exactly n complex roots. This theorem was first proved by Carl Friedrich Gauss in 1799 and is considered one of the most important theorems in mathematics. The theorem states that every polynomial equation of degree n has exactly n complex roots, including real and complex roots.

Understanding Polynomial Functions

A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a constant. The degree of a polynomial function is the highest power of the variable in the function. For example, the polynomial function $f(x) = x^3 - 3x + 1$ is a polynomial function of degree 3.

Applying the Fundamental Theorem of Algebra to the Given Polynomial Function

The given polynomial function is $f(x) = \left(x^3 - 3x + 1\right)^2$ . To determine the number of roots of this function, we need to apply the Fundamental Theorem of Algebra. Since the function is a polynomial function of degree 3, we know that it has exactly 3 complex roots.

However, the function is squared, which means that each root of the original function will be repeated twice. Therefore, the function $f(x) = \left(x^3 - 3x + 1\right)^2$ will have 6 roots, including real and complex roots.

Why Does the Squaring of the Function Affect the Number of Roots?

When a polynomial function is squared, each root of the original function will be repeated twice. This is because the squared function is essentially the product of the original function with itself. Therefore, if the original function has a root at x = a, the squared function will have roots at x = a and x = a.

Conclusion

In conclusion, the Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots. When a polynomial function is squared, each root of the original function will be repeated twice. Therefore, the polynomial function $f(x) = \left(x^3 - 3x + 1\right)^2$ will have 6 roots, including real and complex roots.

Final Answer

The final answer is C. 6 roots.

Additional Information

The Fundamental Theorem of Algebra is a fundamental concept in mathematics that states every non-constant polynomial equation of degree n has exactly n complex roots.
A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a constant.
When a polynomial function is squared, each root of the original function will be repeated twice.
The polynomial function $f(x) = \left(x^3 - 3x + 1\right)^2$ will have 6 roots, including real and complex roots.

References

Gauss, C. F. (1799). Disquisitiones Arithmeticae.
Rudin, W. (1976). Principles of Mathematical Analysis.
Strang, G. (1988). Linear Algebra and Its Applications.
Q&A: Understanding the Fundamental Theorem of Algebra and Polynomial Functions ================================================================================

In our previous article, we explored the concept of the Fundamental Theorem of Algebra and applied it to a given polynomial function to determine the number of roots it has. In this article, we will answer some frequently asked questions related to the Fundamental Theorem of Algebra and polynomial functions.

Q: What is the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra is a theorem in algebra that states every non-constant polynomial equation of degree n has exactly n complex roots. This theorem was first proved by Carl Friedrich Gauss in 1799 and is considered one of the most important theorems in mathematics.

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a variable and a constant. The degree of a polynomial function is the highest power of the variable in the function.

Q: How does the Fundamental Theorem of Algebra apply to polynomial functions?

A: The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n complex roots. This means that if we have a polynomial function of degree n, we can find exactly n complex roots for that function.

Q: What happens when a polynomial function is squared?

A: When a polynomial function is squared, each root of the original function will be repeated twice. This is because the squared function is essentially the product of the original function with itself.

Q: Can you give an example of a polynomial function and its roots?

A: Let's consider the polynomial function $f(x) = x^2 - 4x + 4$ . This function has a degree of 2, which means it has exactly 2 complex roots. To find the roots, we can factor the function as $f(x) = (x - 2)^2$ . This means that the function has a root at x = 2, and this root is repeated twice.

Q: How do we find the roots of a polynomial function?

A: There are several methods to find the roots of a polynomial function, including factoring, the quadratic formula, and numerical methods. The method we use depends on the degree of the polynomial function and the complexity of the roots.

Q: What is the significance of the Fundamental Theorem of Algebra?

A: The Fundamental Theorem of Algebra has far-reaching implications in mathematics and science. It provides a fundamental understanding of polynomial functions and their roots, which is essential in many areas of mathematics, including algebra, geometry, and calculus.

Q: Can you provide some examples of polynomial functions and their roots?

A: Here are a few examples:

$f(x) = x^2 - 4x + 4$ has roots at x = 2 (repeated twice)
$f(x) = x^3 - 6x^2 + 11x - 6$ has roots at x = 1, x = 2, and x = 3
$f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$ has roots at x = 1 (repeated four times)

Q: How does the Fundamental Theorem of Algebra apply to real-world problems?

A: The Fundamental Theorem of Algebra has many applications in real-world problems, including:

Engineering: Polynomial functions are used to model the behavior of physical systems, such as electrical circuits and mechanical systems.
Economics: Polynomial functions are used to model economic systems, including supply and demand curves.
Computer Science: Polynomial functions are used in algorithms and data structures, such as sorting and searching.